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<h4> Data Reduction </h4>
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<p>Data reduction is a step to reduce the size of the subject's functional data. 
Principal Components Analysis (PCA) is used as a technique to reduce the 
dimensions. A single subject is reduced from 53*63*34* 220 to 53*63*34* 50, 
where 220 refers to the time points or scans. </p>
<p>A maximum of three data reduction steps or groups is allowed. First group 
contains the individual data-sets and each data-set is reduced in time dimension 
and put in another group where the data-sets are concatenated into groups and 
put through another data reduction step. Table 1 shows how groups are formed 
depending upon the number of data-sets. Table 2 shows the procedure involved in 
the data-reduction steps. In addition to the tables, figure shows how the 
data-reduction step is performed.</p>
<p class="style1"> <b> Table 1 </b>: Shows how groups are formed depending upon 
the number of data-sets </p>
<table border="2">
  <tr>
    <td width="19%" height="19"><b>Number of Data-Sets (n)</b></td>
    <td width="26%" height="19"><b>Number of groups or reduction steps</b></td>
  </tr>
  <tr>
    <td width="19%" height="19">n = 1</td>
    <td width="26%" height="19">1</td>
  </tr>
  <tr>
    <td width="19%" height="19">n &lt; 4</td>
    <td width="26%" height="19">2</td>
  </tr>
  <tr>
    <td width="19%"  height="19">n >= 4</td>
    <td width="26%" height="19">User specified number (2 or 3, See
    <a href="icatb_setup_ica.htm">Setup ICA</a>)</td>
  </tr>
</table>
<p class="style1"><b>Table 2:</b> Shows how many sub-groups can be formed in a 
group</p>
<table border="2">
  <tr>
    <td width="19%" height="38"> <b> Number of groups or reduction steps </b> </td>
    <td width="81%" height="38"> <b>Procedure</b></td>
  </tr>
  <tr>
    <td width="19%" height="19">1</td>
    <td width="81%" height="19">Only one sub-group can be formed from one 
	data-set. This data-set is reduced in time dimension (scans) to PC1 (See,
    <a href="icatb_setup_ica.htm">Setup ICA</a>).</td>
  </tr>
  <tr>
    <td width="19%" height="76">2</td>
    <td width="81%" height="76">First group contains the number of data-sets. 
	These data-sets are reduced in time dimension to first data-reduction number 
	and put in second group. In the second group, the number of sub-groups (say 
	m) is determined by ratio of number of data-sets divided by 4. The reduced 
	data-sets from first group are placed in m groups and concatenated in each 
	sub-group. Each concatenated sub-group is reduced to a second data-reduction 
	number (PC2, See
    <a href="icatb_setup_ica.htm">Setup ICA</a>).&nbsp;
    </td>
  </tr>
  <tr>
    <td width="19%" height="57">3</td>
    <td width="81%" height="57">First step involves all the steps for two 
	data-reduction steps. Third group contains only one sub-group. In this 
	sub-group, the reduced sub-groups from data-reduction step 2 are 
	concatenated. The concatenated data-set is reduced to a third data-reduction 
	number (PC3, See <a href="icatb_setup_ica.htm">Setup ICA</a>).</td>
  </tr>
</table>
<p class="style1"> <b>Figure 1</b>: Three data-sets reduced in time dimension 
using two data-reduction steps</p>
<p class="style1"><img border="0" src="images/icatb_data_reduction.jpg"></p>
<p><b>Calculation involved in data-reduction step: </b></p>
<p>The example here discussed will be for one data-set. Let us say if the data 
is a matrix of dimensions (V by I) where V is volume and I is images or time 
points. Covariance matrix in time dimension is calculated from the observed data 
and eigen values are computed from the covariance matrix. The eigen values are 
arranged in decreasing order. Only the eigen vectors that correspond to non-zero 
and first m (Reduction parameter PC1) components are taken into account. These 
components are orthogonal to each other and then passed to the whitening step. 
Whitening matrix is computed by solving the square root of diagonal matrix of 
eigen values and transpose of the eigen vectors. This matrix is then multiplied 
to the data to make the variances equal for all the components. Thus the 
components are orthogonal and the variances are equal. </p>
<p><b>Note</b>: The whitening matrix, de-whitening matrix (pseudo inverse of 
whitening matrix) are stored for future use during the
<a href="icatb_back_reconstruct.htm">back reconstruction</a> as they contain 
information regarding the reduced set of the concatenated groups. These are 
stored as .mat files with suffixes as '_pca_r*' where * refers to the reduction 
step count.</p>
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